Holt Algebra 2 7-3 Logarithmic Functions 7-3 Logarithmic Functions Holt Algebra 2 Warm Up Warm Up...

Holt Algebra 2

7-3 Logarithmic Functions7-3 Logarithmic Functions

Holt Algebra 2

Warm Up

Lesson Presentation

Lesson Quiz

Holt Algebra 2

7-3 Logarithmic Functions

Warm Up

Use mental math to evaluate.

1. 4–3

3. 10–5

5. A power has a base of –2 and exponent of 4. Write and evaluate the power.

(–2)4 = 16

2

0.00001

2. 1

416

4.

Holt Algebra 2


Write equivalent forms for exponential and logarithmic functions.

Write, evaluate, and graph logarithmic functions.

Objectives

“I can…”

Holt Algebra 2


logarithm

common logarithm

logarithmic function

Vocabulary

Holt Algebra 2


How many times would you have to double $1 before you had $8? You could use an

exponential equation to model this situation. 1(2x) = 8. You may be able to solve this

equation by using mental math if you know 23 = 8. So you would have to double the

dollar 3 times to have $8.

Holt Algebra 2


How many times would you have to double $1 before you had $512?

You could solve this problem if you could solve 2x = 8 by using an inverse

operation that undoes raising a base to an exponent equation to model this

situation.

This operation is called finding the logarithm. A logarithm is the exponent to

which a specified base is raised to obtain a given value.

Holt Algebra 2


You can write an exponential equation as a logarithmic equation and vice versa.

Read logb

a= x, as “the log base b of a is x.” Notice that the log is the exponent.

Reading Math

Holt Algebra 2


Write each exponential equation in logarithmic form.

Example 1: Converting from Exponential to Logarithmic Form

The base of the exponent becomes the base of the logarithm.

The exponent is the logarithm.

An exponent (or log) can be negative.

The log (and the exponent) can be a variable.

Exponential Equation

Logarithmic Form

35 = 243

25 = 5

104 = 10,000

6–1 =

ab = c

1

6

1

2

log3

243 = 5

1

2log

255 =

log10

10,000 = 4

1

6log

6 = –1

loga

c =b

Holt Algebra 2


Write each exponential equation in logarithmic form.

The base of the exponent becomes the base of the logarithm.

The exponent of the logarithm.

The log (and the exponent) can be a variable.


Logarithmic Form

92= 81

33 = 27

x0 = 1(x ≠ 0)

Check It Out! Example 1

a.

b.

c.

log9

81 = 2

log3

27 = 3

logx1 = 0

Holt Algebra 2


Example 2: Converting from Logarithmic to Exponential Form

Write each logarithmic form in exponential equation.

The base of the logarithm becomes the base of the power.

The logarithm is the exponent.

A logarithm can be a negative number.

Any nonzero base to the zero power is 1.

Logarithmic Form


log99 = 1

log2512 = 9

log82 =

log4 = –2

logb1 = 0

1

16

1

3

91 = 9

29 = 512

1

38 = 2

1

164–2 =

b0 = 1

Holt Algebra 2


Write each logarithmic form in exponential equation.

The base of the logarithm becomes the base of the power.

The logarithm is the exponent.

An logarithm can be negative.

Logarithmic Form


log1010 = 1

log12144 = 2

log 8 = –31

2


101 = 10

122 = 144

1

2

–3

= 8

Holt Algebra 2


A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may

have noticed the following properties in the last example.

Holt Algebra 2


A logarithm with base 10 is called a common logarithm. If no base is written for a

logarithm, the base is assumed to be 10. For example, log 5 = log10

5.

You can use mental math to evaluate some logarithms.

Holt Algebra 2


Evaluate by using mental math.

Example 3A: Evaluating Logarithms by

Using Mental Math

The log is the exponent.

Think: What power of 10 is 0.01?

log 0.01

10? = 0.01

10–2 = 0.01

log 0.01 = –2

Holt Algebra 2



Example 3B: Evaluating Logarithms by

Using Mental Math


Think: What power of 5 is 125?

log

5

125

5? = 125

53 = 125

log

5

125 = 3

Holt Algebra 2



Example 3C: Evaluating Logarithms by Using Mental Math


Think: What power of 5 is ?

log

5

1

5

5? =1

5

5–1 = 1

5

log

5

= –11

5

1

5

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log 0.00001

10? = 0.00001

10–5 = 0.01

log 0.00001 = –5

Check It Out! Example 3a

Holt Algebra 2





log

25

0.04

25? = 0.04

25–1 = 0.04

log

25

0.04 = –1

Check It Out! Example 3b

Holt Algebra 2


Because logarithms are the inverses of exponents, the inverse of an exponential function,

such as y = 2x, is a logarithmic function, such as y = log2

x.

You may notice that the domain and range of each

function are switched.

The domain of y = 2x is all real numbers (R), and the

range is {y|y > 0}. The domain of y = log2

x is {x|x >

0}, and the range is all real numbers (R).

Holt Algebra 2


Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe

the domain and range of the inverse function.

Example 4A: Graphing Logarithmic Functions

f(x) = 1.25x

Graph f(x) = 1.25x by using a table of

values.

1f(x) = 1.25x

210–1–2x

0.64 0.8 1.25 1.5625

Holt Algebra 2


Example 4A Continued

To graph the inverse, f–1(x) = log1.25

x, by using a table of values.

210–1–2f–1(x) = log

1.25x

1.56251.2510.80.64

x

The domain of f–1(x) is {x|x > 0}, and the range is R.

Holt Algebra 2


Example 4B: Graphing Logarithmic Functions

x –2 –1 0 1 2

f(x) =( ) x 4 2 1

Graph f(x) = x by using a table of

values.

1

2

1

2

1

2

1

4

f(x) = x 1

2

Use the x-values {–2, –1, 0, 1, 2}. Graph the function and its inverse. Describe


Holt Algebra 2



To graph the inverse, f–1(x) = log x, by using a table

of values. 1

2

1

2

1

4

1

2

x 4 2 1

f –1(x) =log x –2 –1 0 1 2

Example 4B Continued

Holt Algebra 2



x –2 –1 1 2 3

f(x) = x16

9

4

3

3

4

9

16

27

64

3

4

Use x = –2, –1, 1, 2, and 3 to graph . Then graph its inverse. Describe


Graph by using a table of

values.

Holt Algebra 2



To graph the inverse, f–1(x) = log x,

by using a table of values. 3

4


x

f–1(x) = log x –2 –1 1 2 3

16

9

4

3

3

4

9

16

27

64

3

4

Holt Algebra 2


The key is used to evaluate logarithms in base 10. is used to find

10x, the inverse of log.

Helpful Hint

Holt Algebra 2


The table lists the hydrogen ion concentrations for a number of food items. Find

the pH of each.

Example 5: Food Application

Substance H+ conc. (mol/L)

Milk 0.00000025

Tomatoes 0.0000316

Lemon juice 0.0063

Holt Algebra 2


Milk

Example 5 Continued

The hydrogen ion concentration is 0.00000025 moles per liter.

pH = –log[H+ ]

pH = –log(0.00000025)Substitute the known values in the function.

Milk has the pH of about 6.6.

Use a calculator to find the value of the

logarithm in base 10. Press the key.

Holt Algebra 2


Tomatoes


pH = –log[H+ ]


Tomatoes have the pH of about 4.5.



Example 5 Continued

Holt Algebra 2


Lemon juice


pH = –log[H+ ]


Lemon juice has the pH of about 2.2.



Example 5 Continued

Holt Algebra 2


What is the pH of iced tea with a hydrogen ion concentration of 0.000158 moles

per liter?


pH = –log[H+ ]


Iced tea has the pH of about 3.8.




Holt Algebra 2


Lesson Quiz: Part I

1. Change 64 = 1296 to logarithmic form. log6

1296 = 4

2. Change log27

9 = to exponential form.2

327 = 9

2

3

3. log 100,000

4. log64

8

5. log3

Calculate the following using mental math.

1

27

5

0.5

–3

Holt Algebra 2


6. Use the x-values {–2, –1, 0, 1, 2, 3} to graph f(x) =( )X. Then graph its inverse.

Describe the domain and range of the inverse function.

5

4

Lesson Quiz: Part II

D: {x > 0}; R: all real numbers

Holt Algebra 2 7-3 Logarithmic Functions 7-3 Logarithmic Functions Holt Algebra 2 Warm Up Warm Up...

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